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preconditioning
Sylvain Mailler, François Lott
To cite this version:
Sylvain Mailler, François Lott. Equatorial mountain torques and cold surge preconditioning. Jour- nal of the Atmospheric Sciences, American Meteorological Society, 2010, 67 (6), pp.2101-2120.
�10.1175/2010jas3382.1�. �hal-01136866�
Equatorial Mountain Torques and Cold Surge Preconditioning
S YLVAIN M AILLER
Laboratoire de Me´te´orologie Dynamique du CNRS, Paris, and E ´ cole Nationale des Ponts et Chausse´es, Marne la Valle´e, France
F RANC ¸ OIS L OTT
Laboratoire de Me´te´orologie Dynamique du CNRS, Paris, France
(Manuscript received 17 November 2009, in final form 4 February 2010) ABSTRACT
The evolution of the two components of the equatorial mountain torque (EMT) applied by mountains on the atmosphere is analyzed in the NCEP reanalysis. A strong lagged relationship between the EMT component along the Greenwich axis T
M1and the EMT component along the 908E axis T
M2is found, with a pronounced signal on T
M1followed by a signal of opposite sign on T
M2. It is shown that this result holds for the major massifs (Antarctica, the Tibetan Plateau, the Rockies, and the Andes) if a suitable axis system is used for each of them.
For the midlatitude mountains, this relationship is in part associated with the development of cold surges.
Following these results, two hypotheses are made: (i) the mountain forcing on the atmosphere is well measured by the regional EMTs and (ii) this forcing partly drives the cold surges. To support these, a purely dynamical linear model is proposed: it is written on the sphere, uses an f-plane quasigeostrophic approxi- mation, and includes the mountain forcings. In this model, a positive (negative) peak in T
M1produced by a mountain massif in the Northern (Southern) Hemisphere is due to a large-scale high surface pressure anomaly poleward of the massif. At a later stage, high pressure and low temperature anomalies develop in the lower troposphere east of the mountain, explaining the signal on T
M2and providing the favorable conditions for the cold surge development.
It is concluded that the EMT is a good measure of the dynamical forcing of the atmospheric flow by the mountains and that the poleward forces exerted by mountains on the atmosphere are substantial drivers of the cold surges, at least in their early stage. Therefore, the EMT time series can be an important diagnostic to assess the representation of mountains in general circulation models.
1. Introduction
The influence of mountains on meteorology is a prob- lem of longstanding interest. At the synoptic scales, this interest follows the fact that mountains can help the development of cyclones on their lee side or induce cold surges that can travel equatorward along the eastern mountain flanks. The problem of lee cyclogenesis was studied extensively in the 1980s and 1990s. It has been documented in case studies (e.g., Buzzi and Tibaldi 1978;
Clark 1990) and numerical studies (Egger 1988) and partly explained by theoretical models (Pierrehumbert 1985;
Smith 1984, 1986; Speranza et al. 1985). The cold surges are a growing topic of interest because they are an
important source of atmospheric variability in winter along the eastern flanks of the major mountain ranges (Hsu and Wallace 1985). They are also relevant for the global climate because they connect the meteorology of the midlatitudes to that in the tropics (Garreaud 2001;
Chen et al. 2004). As an illustration of this, the East Asian cold surges interact with the winter monsoon over eastern Asia, the South China Sea, the Maritime Conti- nent, and even the Bay of Bengal (Chang et al. 1979;
Chang and Lau 1980; Slingo 1998; Tangang et al. 2008;
Mailler and Lott 2009).
In mountain meteorology, it is quite common to mea- sure the dynamical influence of mountains on the atmo- sphere by horizontal forces (e.g., Davies and Phillips 1985;
Bessemoulin et al. 1993). In this approach, there is a dis- tinction between the drag forces that are opposed to the low-level winds and the lift forces that are perpendicu- lar to them. At the mesoscale, where the mountain flow dynamics is controlled by gravity waves and includes
Corresponding author address: Franc
xois Lott, Laboratoire deMe´te´orologie Dynamique, Ecole Normale Supe´rieure, 24, rue Lhomond, 75231 Paris CEDEX 05, France.
E-mail: flott@lmd.ens.fr
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DOI: 10.1175/2010JAS3382.1
Ó
2010 American Meteorological Society
low-level flow blocking, the drag force is very important, whereas at the synoptic and planetary scales the lift force plays a key role and causes vortex compression over mountains (Smith 1979). By triggering baroclinic wave development and forcing planetary-scale waves, these lift forces have an impact on the large-scale atmospheric flow and their proper representation in GCMs is an im- portant issue (Lott 1999).
If we adopt a planetary-scale approach, which is man- datory for large-scale mountains, the atmospheric angu- lar momentum (AAM) budget (Feldstein 2006) provides a more satisfactory formalism to analyze these forces, with the mountain forces translating into torques along the three planetary axes. For example, Mailler and Lott (2009) have shown that before convective events asso- ciated with the East Asian winter monsoon, there are strong signals on the equatorial mountain torque (EMT) components T
M1and T
M2due to the Tibetan Plateau (TP). In agreement with other authors (Egger et al. 2007;
Egger and Hoinka 2008), they also show that the positive torques along the Greenwich axis are due to a positive surface pressure anomaly to the north of the TP, while the negative torques along the 908E axis are due to a positive pressure anomaly along its eastern slopes (see Fig. 1, where it is also illustrated how a mountain force F trans- lates into torques along the equatorial axes).
In the past, the analysis of mountain torques was mainly focused on the changes induced in the atmospheric angular momentum and essentially to interpret the changes in the earth orientation parameters (e.g., Rosen and Salstein
1983). In this context, the axial mountain torque T
M3has been analyzed by many authors, including Iskenderian and Salstein (1998), who related the variations of T
M3to the midlatitude dynamics at synoptic scales, and Weickmann and Sardeshmukh (1994), who related its variations to the tropical dynamics at intraseasonal time scales. The two studies cited above are representative of the fact that many authors focused on T
M3, whereas T
M1and T
M2have received less attention. Quite recently, however, Feldstein (2006) has analyzed their relation with changes in the equatorial components of the atmospheric angular mo- mentum, and Egger and Hoinka (2008) have described the regional circulation patterns associated with the equatorial torques produced by the TP.
Beyond understanding the axial AAM budget, the axial mountain torque has also been used to measure the dynamical influence of the mountains on the weather and on the climate. It is in this context that Lott et al. (2004) found that T
M3produces changes in the Arctic Oscilla- tion at periodicities below 30 days. Their results, based on diagnostics from the National Centers for Environmental Prediction (NCEP)–National Center for Atmospheric Research (NCAR) reanalysis, have been confirmed us- ing a GCM (Lott et al. 2005) and explained by a simple comprehensive model (Lott and d’Andrea 2005).
Following these lines of work, the purpose of this paper is to show that the temporal evolution of the two components of the EMT is related to the dynamical forcing of the cold surges by the major midlatitude mountain ranges. Emphasis will be given to the TP and
F
IG. 1. Schematic pictures of midlatitude and subtropical anomalies in surface pressure yielding to anomalous equatorial torques. This
schematic figure is for the Himalayas, and the surface pressure anomalies are representative of two successive stages in the life cycle of an
East Asian cold surge. The vector
Fis a representation of the dominant horizontal force exerted by the mountain on the atmosphere.
the East Asian cold surges, but the case of the Rockies and the Andes will also be discussed. To establish the link between the EMT and the cold surges, we will use two pieces of independent evidence. The first is based on a statistical analysis of the NCEP reanalysis data for which the two components of the EMT are evaluated globally and for each major mountain range separately.
The temporal evolution of the surface patterns of pressure and temperature associated with the two EMT compo- nents are then analyzed and compared to those occurring during cold surges. The second is based on an analytical model that gives a comprehensive explanation for the relationship between the EMT components and the cold surges. This model is purely dynamical and only retains two essential features of the cold surge dynamics: the cold surges are related to low-level cold advection from the midlatitudes to the tropics, and they are preceded by high pressure anomalies poleward of the mountain massif. Because of its simplicity, this model is only used to show that mountain forcings, yielding to EMTs with amplitudes comparable with those observed, are largely sufficient to trigger the initial phase of cold surges.
The plan of the paper is as follows. Section 2 presents the statistical analysis. From the NCEP–NCAR reanalysis, we evaluate the two components of the EMT globally and for each major mountain range. For the Rockies, the Andes, and the TP we then perform a composite analysis
of the surface fields keyed to the EMT. Section 3 pres- ents the theoretical model and our interpretation for the relation between the mountain forcing and the EMTs.
Section 4 uses the model results to interpret the obser- vational results of section 2. Section 5 summarizes and discusses the significance of our results for (i) the cold surge preconditioning, (ii) the equatorial atmospheric angular momentum budget, and (iii) the representation of mountains in GCMs.
2. Statistical analysis
Our data are taken from the NCEP–NCAR reanalysis (Kalnay et al. 1996) products for the 1979–2007 period.
Specifically, we use the surface pressure P
S, the 0.995 sigma level temperature T
50m, and the surface elevation h.
We will also use the tridimensional fields of zonal wind u and temperature T. To focus on the subseasonal vari- ability, we will often remove the annual cycle by sub- tracting the daily climatology obtained by averaging over the 29 years of our dataset. When used, those subseasonal data will be identified by a tilde symbol; for instance, the subseasonal surface pressure will be written P e
s.
a. Spectral analysis of the equatorial mountain torques
The two components of the global subseasonal EMT acting on the atmosphere are computed following Feldstein (2006):
e
T
M1(t) 5 a
2ð ð
f,l
h(l, f) sinl cosf › P e
s›f 1 cosl sinf › P e
s›l
!
df dl, (1a)
e
T
M2(t) 5 a
2ð ð
f,l
h(l, f) cosl cosf › P e
s›f 1 sinl sinf › P e
s›l
!
df dl, (1b)
where a, f, and l are the earth’s radius, latitude and longitude, respectively. In (1), T e
M1is the component of
the EMT along the equatorial axis, which is at 08 longi- tude (the Greenwich axis), and T e
M2is the component
T
ABLE1. Domains over which the integrations in (1a) and (1b) are evaluated to measure the individual contributions of each mountain massif to the global torque, and geographical coordinates of the center of mass for each mountain range. Note that the do- main sizes here largely contain the massifs considered.
Massif
Longitude range [l
X1;
lX2]
Latitude range [f
X1;
fX2]
Center of mass (l
X,
fX) Andes [908–608W] [608S–158N] (16.78S, 70.48W) Antarctica [1808W–1808E] [908–608S] (83.18S, 79.78E) Greenland [708–108W] [558–858N] (72.08N, 43.68W) Tibetan Plateau [608–1208E] [158–608N] (37.88N, 91.58E) Rockies [1708–908W] [158–808N] (44.98N, 111.38W)
T
ABLE2. Standard deviation
hT
eX12 1T
eX22 i1/2for the global EMT and for the regional contributions of each of the major mountain ranges. Units are Hadleys.
Massif All-year Winter (DJF) Summer (JJA)
Global 63.6 60.3 65.2
Antarctica 50.2 36.5 58.1
Tibetan Plateau 21.8 26.8 14.9
Greenland 16.8 21.1 11.2
Rockies 15.4 19.1 10.3
Andes 6.0 7.6 4.3
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along the equatorial axis at 908E. Following Egger and Hoinka (2000), we will express all torques in Hadleys (Ha) with 1 Ha [ 10
18kg m
2s
22.
The individual contributions of the major mountain ranges (here the TP, the Rockies, the Andes, Antarctica, and Greenland) to the EMT can be computed by re- stricting the integrations in (1) to sectors including the considered mountain range, which are given in Table 1.
Nevertheless, to compare the dynamics associated with the EMT produced by these mountains, the fact that they are located at different places has to be taken into ac- count. For instance, that a northward force applied to the atmosphere around the central location of the TP will mainly result in a positive T e
M1(and a small T e
M2) is due to the fact that the TP is centered around the longitude 908E (Table 1). In the following we will note this longitude l
TPand also note l
Xand f
X, the longitude and the latitude of the center of mass of the mountain massif X (with X 5 TP, R, Ad, At, and G for the TP, Rockies, Andes,
Antarctica, and Greenland, respectively). More pre- cisely, they are the longitude and latitude of the vector
r
X5 ð
l2Xl1X
ð
f2X f1Xh(l, f)r(l, f) cosf df dl ð
l2Xl1X
ð
f2X f1Xh(l, f) cosf df dl
, (2)
where r(l, f) 5 a(cosl cosf, sinl cosf, sinf), and l
1X, l
2X, f
1X, and f
2Xare the limits of a region that largely contains the mountain massif considered. These limits as well as l
Xand the f
Xare given in Table 1.
To make the results for different mountains compa- rable, the local equatorial torques for a mountain range X will be computed along equatorial axes rotated by l
X2 p/2, which means that the second equatorial axis is now at the longitude l
Xof the mountain considered. As said before, this rotation is not needed for the TP and is not adapted for Antarctica, which covers all longitudes.
F
IG. 2. Spectral analysis of T
eM1and T
eM2: (a) coherency between T
eM1and T
eM2, (b) phase difference between T
eM1and T
eM2, (c) spectrum of T
eM1, and (d) spectrum of T
eM2. The spectra and the cross-spectrum are deduced from the two periodograms and the cross periodogram of the series, each of them being smoothed by an 80-point 10% cosine window. In the frequency domain this yields a resolution of 8
310
22day
21. The mean level (gray solid) and the 99%
significance level (gray dashed) are evaluated by a Monte Carlo procedure, which uses an ensemble of 500 pairs of red
noise series whose variances and lag-1 autocorrelations correspond to those of T
eM1and T
eM2.
F
IG. 3. Coherence and phase relationship between (a) T
eTP1and T
eTP2(b) T
eR1and T
eR2(c) T
eAd1and T
eAd2and (d) T
eAt1and T
eAt2. Conventions and method are as in Fig. 2.
F
IG. 4. (a) Composite time series of T
eTP1(gray, solid) and T
eTP2(black, solid) keyed on the 20 strongest positive peaks of T
eTP1and the corresponding 99% significance levels. (b) As in (a) but for T
eR1and T
eR2keyed on T
eR1. (c) Composite time series of T
eAd1and T
eAd2keyed on the 20 strongest negative peaks of T
eAd1.
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In the following, the subseasonal regional contributions to the EMTs expressed in these rotated axes will be denoted ( T e
X1; T e
X2), where X is the name of the corre- sponding mountain massif. As an illustration, T e
X15 0 and T e
X2. 0 means that the contribution of the massif X to the EMT vector is oriented along the longitude l
X. The variance of each of the contributions is given in Table 2, indicating that, while Antarctica is the strongest single contributor, the midlatitude massifs can have a sig- nificant impact, at least during the Northern Hemisphere winter.
To characterize the coherence and phase relation- ship between the two components of the global EMT, we next perform a conventional cross-spectral analysis between the corresponding series (Fig. 2, see caption for details on the method). The cospectrum between T e
M1and T e
M2in Fig. 2a shows that the two torques are sig- nificantly related with each other for periodicities be- tween 2 and 20 days, a spectral domain that contains most of the variance of T e
M1and T e
M2(see Figs. 2c,d). The phase in Fig. 2b of nearly p/2 shows that T e
M1and T e
M2are close to being in quadrature, with the EMT vector ro- tating westward.
The same analysis applied to the rotated EMTs asso- ciated with the individual mountains in Fig. 3 shows that for the TP, the Rockies, the Andes, and Antarctica, T e
X1and T e
X2are close to being in quadrature as well (Fig. 3), which means that the regional EMT vectors also rotate westward. These results are consistent with those in Egger and Hoinka (2008), who considered the TP only and used lagged correlations rather than a spectral analysis.
The fact that all the regional EMT vectors (except Greenland, not shown) rotate westward like the global EMT indicates that the global behavior of the EMT is, in part, the result of regional circulations, the global EMT vector being the sum of all the regional contributions.
b. Regional circulation patterns
One of the central points we wish to make is that, when a high pressure anomaly is located poleward of a given mountain massif X in the northern (southern) midlatitudes (i.e., X 5 TP, R, or Ad), it gives rise to a positive (negative) mountain torque T e
X1, and this mountain torque in part controls the future evolution of the flow, triggering a cold surge and a negative (positive)
e
T
X2. To support this picture observationally, composites will be built for the surface fields keyed to the positive peaks of T e
TP1and T e
R1and to the negative peaks of T e
Ad1. In the following, the time lags are expressed in days relative to the peaks in T e
X1: D0 is the day when T e
X1peaks, D 2 2 is two days before, and D 1 2 is two days after. For instance, in the case of the TP at D0, the composite is the average of the values found for the days
when T e
TP1exceeds a given positive threshold. The threshold is chosen so that 20 positive peaks in T e
TP1are selected, and we impose a 20-day gap between the dates selected. This last restriction, which ensures a statistical independence between the dates selected, allows one to use a Student’s t test to evaluate the confidence levels.
At D 1 l (D 2 l), the composite is built with the values corresponding to the dates situated at the lag l days after (before) the 20 positive peaks identified above.
Figure 4 shows the composites of T e
X2keyed to T e
X1for the TP, the Rockies, and the Andes. In it, we see that for these three massifs, the peaks in T e
X1are followed by peaks of the opposite sign in T e
X2. Note that a less significant signal in T e
X2of the same sign as T e
X1tends to precede. This result is consistent with the cross-spectral analysis in Figs. 3a–c with T e
X1and T e
X2in quadrature, but provides the additional information that the signal in
e
T
X2is larger after than before the peaks in T e
X1. This is a first indicator that T e
X1contributes to the establishment of T e
X2at a later stage. From a more global viewpoint, we have also analyzed the composites of the global EMT keyed to the regional ones (not shown). In agreement with the results in Table 2, we found that the peaks in the EMTs due to the Rockies and the TP are related to sig- nificant signals on both components of the global EMT.
1) T IBETAN P LATEAU
The composites of P e
Sand T e
50mkeyed on the maxima of T e
TP1in Fig. 5 show that the anomalies of T e
TP1and
e
T
TP2are linked to the clockwise movement of cold temperature and high pressure anomalies around the TP, from Siberia to eastern China. This movement begins at D 2 2 (Figs. 5a,b) with the formation of high pressure and low temperature anomalies centered in western Siberia, later strengthening and moving to central Siberia at D0 (Figs. 5c,d). These anomalies are strong (16 hPa and 10 K) and largely significant. The strong anomaly of surface pressure north of the TP at D0 corresponds to the situation presented in Fig. 1a and explains the peak of T e
TP1at that time (Fig. 4a).
At D 1 2, the surface pressure and surface tempera- ture anomalies have shifted to eastern China (Figs. 5e,f).
At this time, the pressure gradient across the TP is mostly in the zonal direction, corresponding to the schematic picture in Fig. 1b, and this induces a negative T e
TP2(Fig.
4a). The surface pressure and surface temperature pat- terns at that time (Figs. 5e,f) are strongly reminiscent of East Asian cold surges (see Fig. 7 of Zhang et al.
1997). Finally, at D 1 4, the anomaly in surface pressure
weakens and moves toward the South China Sea. The
anomaly in surface temperature remains strong (7 K in
southeastern China) and moves to the tropics (up to 4 K
in the northern part of the South China Sea), as occurs at
F
IG. 5. Composite of the (left) surface pressure anomalies and (right) surface temperature anomalies for the 20 strongest positive peaks of T
eTP1at lag (a),(b) D
22, (c),(d) D0, (e),(f) D
12, and (g),(h) D
14; 99% confidence level (shaded). Contour interval is 2 hPa for the surface pressures and 2 K for the surface temperatures: zero contours omitted.
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the end of the life cycle of a cold surge (Zhang et al. 1997).
It is worth noting that the anomalies associated with peaks in T e
TP1begin their life cycle in the Siberian mid- latitudes and finish it 4–6 days later in tropical East Asia.
2) R OCKIES
The composites of P e
Sand T e
50mkeyed on the maxima of T e
R1are shown in Fig. 6. These composites are quite similar to those for the TP in Fig. 5: anomalies of high P e
Sand low T e
50mare located over Alaska and the Arctic Ocean at D 2 2 (Figs. 6a,b) and move to the Canadian plains at D0 (Figs. 6c,d), being very strong at this time
(22 hPa and 12 K at D0). At D 1 2 (Figs. 6e,f), they stretch southeastward along the eastern flank of the Rockies, covering the Gulf of Mexico and a part of central America. The anomalies displayed at that time are very similar to the structure of the North American cold surges, with cold air masses traveling southward from Canada to the Gulf of Mexico (Colle and Mass 1995) and occasionally reaching the eastern Pacific across Central America (Schultz et al. 1997). At D 1 4 and after, the composites lose significance more rapidly than in the case of the TP (not shown). The shift of the high pressure anomaly from the northwestern slopes
F
IG. 6. As in Fig. 5 but keyed on the positive peaks of T
eR1.
F
IG. 7. As in Fig. 5 but keyed on the negative peaks of T
eAd1.
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of the Rockies to their eastern slopes between D 2 2 and D 1 2 explains that T e
R2goes from positive to negative values during this period (Fig. 4b).
3) A NDES
The composites of P e
Sand T e
50mkeyed on the minima of T e
Ad1are in Fig. 7. At D 2 2 (Fig. 7a), a high pressure system is arriving from the Pacific to the southern tip of the Andes. At D0 in Figs. 7c,d, this anticyclone begins to interact substantially with the Andes, with a strong ridge building up along the eastern flank of the mountains.
This ridge is associated with a cold anomaly extending meridionally from 508 to 208S along the eastern flank of the Andes. At D 1 2 the signal on the eastern flank of the Andes becomes even more pronounced (reaching 10 hPa and 210 K around 258S). These strong anoma- lies at subtropical latitudes are quite remarkable and correspond to the ones generated by the South Ameri- can cold surges (Seluchi and Nery 1992; Marengo et al.
1997; Garreaud 2000), suggesting that negative events on T e
Ad1are an important factor in the generation of these events. Similarly to what happens in the case of the TP (Fig. 5), the low temperature and high surface pres- sure anomaly have moved farther into the tropics at D 1 4 (Fig. 7g), being significant as far north as 58N. The evo- lution of the surface pressure patterns explains the evo- lution of the composite T e
Ad2as shown in Fig. 4c: T e
Ad2has a negative peak at D 2 2 when the high pressure arrives from the Pacific Ocean and reaches the western slopes of the Andes (Fig. 7a), then T e
Ad2has a positive tendency at D0 owing to the ridge formation along the eastern moun- tain slopes (Fig. 7c) and reaches strong positive values at D 1 1 and D 1 2 when the ridge further strengthens and moves northward along the mountain (Fig. 7e).
3. Theoretical model a. Basic equations
To interpret the temporal evolution of the EMT vector and its relation with the surface fields discussed in section 2, we will next use a linear model based on the quasi- geostrophic (QG) f-plane approximation of the anelastic equations on the sphere. In this model, we will impose a background zonal geostrophic wind u
g(f, z) in thermal wind balance with a background potential temperature u(f, z),
f u
gz5 g
au
0u
f, (3)
with the Coriolis parameter f 5 2V sinf
r, where f
ris a constant reference latitude. Still in (3), u
0(z) is a background potential temperature corresponding to the
atmosphere at rest. In this linear framework, the equa- tions for the disturbance produced by the mountain are
›
›t 1 u
ga cosf
›
›l
u9
g1 (u
gcosf)
fa cosf y9
gfy9 5 1
a cosf
›F9
›l , (4a)
›
›t 1 u
ga cosf
›
›l
y9
g1 2 u
gu9
ga tanf 1 fu9 5 1
a
›F9
›f 1 F9
r
rd(z), (4b)
›F9
›z g u9
u
05 0, (4c)
r
0a cosf
›u9
›l 1 ›y9 cosf
›f
1 ›r
0w9
›z 5 0, (4d)
›
›t 1 u
ga cosf
›
›l
u9 1 u
fa y9
g1 du
0dz w9 5 Q9(l, f) u
0zr
0, (4e)
and the linear lower boundary condition is w9(z 5 0) 5 u
ga cosf
›h
›l . (5)
In (4a–e) and (5), the prime variables are for the dis- turbance fields produced by the mountain; r
0(z) is a background density profile; r
r5 r
0(z 5 0); u
gand y
gare the two components of the geostrophic wind, where
u
g5 1 af
›F
›f ; y
g5 1 af cosf
›F
›l ; (6) u, y, and w are the three components of the wind; and u is the potential temperature.
Two unconventional terms are introduced in (4b) and (4e). The first is the lateral force F9d(z)/r
rin which d(z) is the Dirac distribution. It will only be used in a formal way to clarify the relation between the mountain forces applied to the atmosphere and the more conventional boundary condition in (5). In (4e) the diabatic term Q9(l, f)u
0z/r
0will be used to represent more realistically the interaction between the low-level wind and the mountain (see section 3).
b. Model description
To analyze the response in our model, we will first
follow Bretherton (1966) and assume that (4a)–(4e),
which are written for z $ 0, are valid for all z once multiplied by the Heaviside function H(z). Hence, if we derive a potential vorticity (PV) budget from (4a)–(4e), we obtain the PV equation
›
›t 1 u
ga cosf
›
›l
q9
g1 y9
ga q
gfH(z)
1 ›
›t 1 u
ga cosf
›
›l
f F9
zN
2f
2u
gzN
2y9
g1 fw9
"
F9
lr
ra cosf fQ9 2u
0z# d(z) 5 0,
(7)
where d(z) 5 H
z(z). Equation (7) can only be satisfied if the terms within the two pairs of brackets are null separately. In the first bracket of (7) the inflow QG PV is given by
q
g5 y
gl(u
gcosf)
fa cosf 1 f
r
0r
0F
zN
2z
. (8)
To represent a high surface pressure anomaly pole- ward of the mountain, we impose a negative background low-level wind u
g(f
r, z 5 0), 0. To represent the sub- tropic to pole negative temperature gradient, we will consider that the background wind has a positive vertical wind shear characterized by the constant L . 0,
u
g(f, z) 5 (u
r1 Lz) cosf, (9) where u
r, 0 is a constant and the cosf dependence is introduced because it simplifies considerably the lower boundary condition given in the second bracket of (7).
The zonal wind in (9) has a mean QG PV q
g5 (u
gcosf)
fa cosf 1 f (r
0F
z)
zN
2r
05 2 u
r1 Lz
a 1 af
2L N
2H
!
sinf, (10)
which varies in the meridional direction. Nevertheless, its meridional gradient q
gf/a compares in amplitude with the planetary PV gradient. As the planetary PV gradient is not taken into account in the f-plane approximation we make, we will consistently neglect the influence of q
gfon the evolution of the disturbances. With this approxi- mation, the disturbance inflow PV q9
gin the first bracket of (7) stays null if it is null at t 5 0, yielding
DF9 1 f
2r
0r
0F9
zN
2z
5 0, (11)
where
D 5 1 a
2cosf
›
›f cosf ›
›f 1 1 a
2cos
2f
›
2›l
2.
Using the kinematic boundary condition (5), the lower boundary condition in the second bracket of (7) becomes
›
t1 u
ra ›
lF9
zL a
›F9
›l 5 N
2u
ra
›h
›l , (12) where we have set the unconventional terms F9 5 Q9 5 0.
The solutions to (11) and (12) can be computed using the spherical harmonic functions Y
lm(l, f), where DY
lm5 2a
22l(l 1 1)Y
lmand ›
lY
lm5 imY
lm. In this formalism, we can project F9 and h on the functions Y
lm,
F9(l, f, z, t) 5 åL
l51
å
1lm5l
F
ml(t)e
klzY
ml(l, f), (13a)
h(l, f) 5 å
L
l51
å
1l
m5l
h
ml(t)Y
ml(l, f), (13b) where L is the truncation and
k
l5 1 4H
21 N
2f
2l(l 1 1)
a
21/2
1
2H . 0. (14) The exponential term in (13a) ensures that the distur- bance has 0 PV and vanishes at z 5 ‘. If we take as the initial condition that the flow at t 5 0 is undisturbed (F
lm5 0 at t 5 0), then a solution that satisfies the linear boundary condition in (12) is
F
ml(t) 5 1 exp imu
rak
l(k
l1 L/u
r)t
(k
l1 L/u
r)
3 N
2h
ml. (15)
c. Equatorial mountain torques in the model
We can calculate two different contributions to the equatorial torques from our model. The first is the one associated with the background surface pressure gradient,
f u
g(0) 5 1 a
›F
s›f 5 1 r
r1 a
›P
s›f , (16) that is present at the initial time in our model. If we take P
sfor the pressure in the equatorial torques in (1a) and (1b), we obtain at t 5 0
T ^
X15a
3f r
rðð
sinl cosfu
g(f)h(l, f) dl df
’af r
ru
g(f
r)V . 0(NH), (17a)
J
UNE2010 M A I L L E R A N D L O T T 2111
T ^
X25 a
3f r
rð ð
cosl cosfu
g(f)h(l, f)dl df ’ 0, (17b) where the caret indicates that these torques are issued from the model. For each massif, they are expressed in the rotated axis system used to define the regional torques in section 2. The approximate values to the right of (17a) and (17b), where V is the mountain volume, are given to indicate that at t 5 0 the EMT is essentially along the first rotated equatorial axis, simply because sinl ’ 1 and cosl ’ 0 where the mountain is located and according to the rotated axes. The sign for T ^
X1is related to the fact that u
r, 0 and to the sign of the Coriolis parameter, so it is positive in the Northern Hemisphere [NH in (17a)] and negative in the Southern Hemisphere.
In this paper we argue that those torques measure the dynamical forcing of the mountain on the flow. This assumption is not so obvious at this stage since the
evaluation of the torque in (1) is just a diagnostic where a given surface pressure field is placed in given integrals.
To illustrate how this torque impacts the future evolution of the flow, it is worth noting that the model equation (7) remains unchanged if we impose h 5 0 in the boundary condition (5) but take the topography into account by the introduction of a surface stress:
w9(z 5 0) 5 0, F9 5 r
rf u
g(f, z 5 0)h. (18) Note also that the stress F9 in (18) corresponds to the lift force associated with vortex compression by mountains in Smith (1979), which was introduced in a GCM by Lott (1999) to reduce its systematic errors.
To translate F9 into a torque, we form the cross pro- duct between the position vector r and the momentum equations (4a) and (4b) and integrate over all the atmo- sphere. When doing so, the EMT vector due to the surface stress F9 is
F
IG. 8. (a) Topography retained for the model run and (b) NCEP topography (m MSL). The model topography is obtained by multiplying the T21-filtered NCEP topography by
a(l)b(f), wherea(l) [b(f)] is a function whose valueis 1 in the interval (l
1;
l2) [(f
1;
f2)] corresponding to the longitude (resp. latitude) range in Table 1; 0 outside (l
12158;
l21158) [(f
12158;
f21158)]; and a smooth connection is ensured by the use of an exponential taper
u(t)5exp[t
2/(t
221)].
F
IG. 9. (a) Composite of the zonal wind at 308N keyed on the 20 strongest positive peaks of T
eTP1; (b) as in (a) but for the composite keyed
on T
eR1; (c) composite of the zonal wind at 308S keyed on the 20 strongest negative peaks of T
eAd1. The contours are every 10 m s
21for
positive (westerly) winds and at
23 and25 m s21for negative (easterly) winds.
T
F95 a
3ðð
l,f